Related papers: Runge-Kutta methods for third order weak approxima…

We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that under certain…

Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid order reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes are…

The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by…

In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…

Measurement noise is an integral part while collecting data of a physical process. Thus, noise removal is a necessary step to draw conclusions from these data, and it often becomes quite essential to construct dynamical models using these…

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high…

In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size,…

A focus of recent research in quantum computing has been on developing quantum algorithms for differential equations solving using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which…

A class of explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (GEPTRKN) methods for solving second-order initial value problems $y'' = f(t,y,y')$, $y(t_0) = y_0$, $y'(t_0)=y'_0$ has been studied. This new class of methods can be considered a…

Symplectic partitioned Runge--Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the…

We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal…

We develop continuous-stage Runge-Kutta-Nystr\"{o}m (csRKN) methods for solving second order ordinary differential equations (ODEs) in this paper. The second order ODEs are commonly encountered in various fields and some of them can be…

In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its…

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…

This work generalizes the additively partitioned Runge-Kutta methods by allowing for different stage values as arguments of different components of the right hand side. An order conditions theory is developed for the new family of…

High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an…

We study the convergence of a class of Runge-Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability…

Extended Stability Runge-Kutta (ESRK) methods are crucial for solving large-scale computational problems in science and engineering, including weather forecasting, aerodynamic analysis, and complex biological modelling. However, balancing…

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schr\"odinger equation by combining the scalar auxiliary variable approach with the exponential…